Functiones et Approximatio Commentarii Mathematici

Bounding the least prime ideal in the Chebotarev Density Theorem

Asif Zaman

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Let $K$ be a number field and suppose $L/K$ is a finite Galois extension. We establish a bound for the least prime ideal occurring in the Chebotarev Density Theorem. Namely, for every conjugacy class $C$ of $\mathrm{Gal}(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ unramified in $L$, for which its Artin symbol $\big[ \frac{L/K}{\mathfrak{p}} \big] = C$, for which its norm $N^K_{\mathbb{Q}}\mathfrak{p}$ is a rational prime, and which satisfies \[ N^K_{\mathbb{Q}} \mathfrak{p} \ll d_L^{40}, \] where $d_L = |\mathrm{disc}(L/\mathbb{Q})|$. All implicit constants are effective and absolute.

Article information

Funct. Approx. Comment. Math., Volume 57, Number 1 (2017), 115-142.

First available in Project Euclid: 28 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R44: Distribution of prime ideals [See also 11N05]
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

least prime ideal Chebotarev Density Theorem Dedekind zeta function Deuring-Heilbronn phenomenon


Zaman, Asif. Bounding the least prime ideal in the Chebotarev Density Theorem. Funct. Approx. Comment. Math. 57 (2017), no. 1, 115--142. doi:10.7169/facm/1651.

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